Question

4. In 1974-76, Stephen Hawking found a remarkable relationship between the entropy of black holes and their surface area. For a non-rotating, uncharged black hole, the precise relationship is 4Gh 4tRS is the area of the black hole of mass M with Schwarzschild radius R -2GM /c2 where A (a) Show that the temperature of the black hole is given by -- Thc [2 marks] (b) If the black hole radiates according to the Stefan-Boltzmann law, show that the rate of change of black hole mass is given by 4 dM dt 15360TG M2 [4 marks] (c) For an initial black hole mass Mo, show that the black hole evaporation time is 5120TG?M) evap 4 [4 marks] [Hint: In Q4 you may find E-Mc useful]

Answer 1

(3-The relation betueen'the entr oet of Lloa. holes and thein Sunface airea is iven StePlen Hauwkng Kis ltzmano con sout here ASunface area .is mass of e block hole. Using the thenmodynamic em Penadune nelation now use Einstein'snelationEM OS h c whene -Ve sign is due to the decrnle in enen g E of deblack boely of area A at temperature

12 15360(? 6几亿ade oS change of-the lefa( hote mass is 5 360 M ew tew Me 2 on tevap eVaup